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Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\\mid q^{n}-1$ . We say that the extension $\\mathbb{F}_{q^{n}}/\\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements property if, for every $\\unicode[STIX]{x1D6FC},\\unicode[STIX]{x1D703}\\in \\mathbb{F}_{q^{n}}^{\\ast }$ such that $\\mathbb{F}_{q^{n}}=\\mathbb{F}_{q}(\\unicode[STIX]{x1D703})$ , there exists some $x\\in \\mathbb{F}_{q}$ such that $\\unicode[STIX]{x1D6FC}(\\unicode[STIX]{x1D703}+x)$ has multiplicative order $(q^{n}-1)/r$ . We prove that, for sufficiently large prime powers $q$ , $\\mathbb{F}_{q^{n}}/\\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements. We also discuss the (weaker) translate property for extensions.

Given $A\\subseteq GL_2(\\mathbb {F}_q)$ , we prove that there exist disjoint subsets $B, C\\subseteq A$ such that $A = B \\sqcup C$ and their additive and multiplicative energies satisfying $$\\begin{align*}\\max\\{\\,E_{+}(B),\\, E_{\\times}(C)\\,\\}\\ll \\frac{|A|^3}{M(|A|)}, \\end{align*}$$ where $$ \\begin{align*} M(|A|) = \\min\\Bigg\\{\\,\\frac{q^{4/3}}{|A|^{1/3}(\\log|A|)^{2/3}},\\, \\frac{|A|^{4/5}}{q^{13/5}(\\log|A|)^{27/10}}\\,\\Bigg\\}. \\end{align*} $$ We also study some related questions on moderate expanders over matrix rings, namely, for $A, B, C\\subseteq GL_2(\\mathbb {F}_q)$ , we have $$\\begin{align*}|AB+C|, ~|(A+B)C|\\gg q^4,\\end{align*}$$ whenever $|A||B||C|\\gg q^{10 + 1/2}$ . These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh ([2019], Expanding phenomena over matrix rings, $Forum Math.$ , 31, 951–970).

In this paper we propose an efficient multivariate encryption scheme based on permutation polynomials over finite fields. We single out a commutative group ℒ( ) of permutation polynomials over the finite field . We construct a trapdoor function for the cryptosystem using polynomials in ℒ(2, ), where =2 for some ≥ 0. The complexity of encryption in our public key cryptosystem is ) multiplications which is equivalent to other multivariate public key cryptosystems. For decryption only left cyclic shifts, permutation of bits and xor operations are used. It uses at most 5 +3 – 4 left cyclic shifts, 5 +3 + 4 xor operations and 7 permutations on bits for decryption.

In this paper we consider a family F of 2 n -dimensional F q -linear rank metric codes in F q n × n arising from polynomials of the form x q s + δ x q n 2 + s ∈ F q n [ x ] . The family F was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8 , and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n . In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n . In this paper we deal with the open case n = 8 , providing a classification for any large enough odd prime power q . The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F q -rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far.

For each odd prime power q , we present two rational functions f ( X ) ∈ F q ( X ) which have the unusual property that, for every odd n , the function induced by f ( X ) on F q n \\ F q is ( q - 1 ) -to-1.

Ovoids of the parabolic quadric Q (6,  q ) of PG ( 6 , q ) have been largely studied in the last 40 years. They can only occur if q is an odd prime power and there are two known families of ovoids of Q (6,  q ), the Thas-Kantor ovoids and the Ree-Tits ovoids, both for q a power of 3. It is well known that to any ovoid of Q (6,  q ) two polynomials f 1 ( X , Y , Z ) , f 2 ( X , Y , Z ) can be associated. In this paper we classify ovoids of Q (6,  q ) with max { deg ( f 1 ) , deg ( f 2 ) } < ( 1 6.3 q ) 3 13 - 1 .

Can any element in a sufficiently large finite field be represented as a sum of two dth powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal equations. Then, we offer two proofs, one new and elementary, and the other more classical, based on Fourier analysis and an application of a nontrivial estimate from the theory of finite fields. In context and juxtaposition, each will have its merits.

We give an elementary proof of Warning's second theorem on the number of solutions to the system of polynomial equations over finite fields.

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on extensions of finite fields, especially for the cases of quadratic and cubic extensions. Our achievements are obtained by investigating absolute irreducibility of some polynomials in two indeterminates.